256 research outputs found
Polymeric Phase of Simplicial Quantum Gravity
We deduce the appearance of a polymeric phase in 4-dimensional simplicial
quantum gravity by varying the values of the coupling constants and discuss the
geometric structure of the phase in terms of ergodic moves. A similar result is
true in 3-dimensions.Comment: 6 pages, revte
Large deviation principles for nongradient weakly asymmetric stochastic lattice gases
We consider a lattice gas on the discrete d-dimensional torus
with a generic translation invariant, finite range
interaction satisfying a uniform strong mixing condition. The lattice gas
performs a Kawasaki dynamics in the presence of a weak external field E/N. We
show that, under diffusive rescaling, the hydrodynamic behavior of the lattice
gas is described by a nonlinear driven diffusion equation. We then prove the
associated dynamical large deviation principle. Under suitable assumptions on
the external field (e.g., E constant), we finally analyze the variational
problem defining the quasi-potential and characterize the optimal exit
trajectory. From these results we deduce the asymptotic behavior of the
stationary measures of the stochastic lattice gas, which are not explicitly
known. In particular, when the external field E is constant, we prove a
stationary large deviation principle for the empirical density and show that
the rate function does not depend on E.Comment: Published in at http://dx.doi.org/10.1214/11-AAP805 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large deviations of the empirical flow for continuous time Markov chains
We consider a continuous time Markov chain on a countable state space and
prove a joint large deviation principle for the empirical measure and the
empirical flow, which accounts for the total number of jumps between pairs of
states. We give a direct proof using tilting and an indirect one by contraction
from the empirical process.Comment: Minor revision, to appear on Annales de l'Institut Henri Poincare (B)
Probability and Statistic
Dynamical Phase Transitions for Fluxes of Mass on Finite Graphs
We study the time-averaged flux in a model of particles that randomly hop on
a finite directed graph. In the limit as the number of particles and the time
window go to infinity but the graph remains finite, the large-deviation rate
functional of the average flux is given by a variational formulation involving
paths of the density and flux. We give sufficient conditions under which the
large deviations of a given time averaged flux is determined by paths that are
constant in time. We then consider a class of models on a discrete ring for
which it is possible to show that a better strategy is obtained producing a
time-dependent path. This phenomenon, called a dynamical phase transition, is
known to occur for some particle systems in the hydrodynamic scaling limit,
which is thus extended to the setting of a finite graph
Large deviations for a stochastic model of heat flow
We investigate a one dimensional chain of harmonic oscillators in which
neighboring sites have their energies redistributed randomly. The sites
and are in contact with thermal reservoirs at different temperature
and . Kipnis, Marchioro, and Presutti \cite{KMP} proved that
this model satisfies {}Fourier's law and that in the hydrodynamical scaling
limit, when , the stationary state has a linear energy density
profile , . We derive the large deviation
function for the probability of finding, in the stationary
state, a profile different from . The function
has striking similarities to, but also large differences from, the
corresponding one of the symmetric exclusion process. Like the latter it is
nonlocal and satisfies a variational equation. Unlike the latter it is not
convex and the Gaussian normal fluctuations are enhanced rather than suppressed
compared to the local equilibrium state. We also briefly discuss more general
model and find the features common in these two and other models whose
is known.Comment: 28 pages, 0 figure
Box-ball system: soliton and tree decomposition of excursions
We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514–3519, 1990). Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari et al. (Soliton decomposition of the box-ball system (2018). arXiv:1806.02798) proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020) propose a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In the present paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall (Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85, vol. 1204, pp. 447–464. Lecture Notes in Mathematics. Springer, Berlin (1986)). A ball configuration distributed as independent Bernoulli variables of parameter λ < 1∕2 is in correspondence with a simple random walk with negative drift 2λ − 1 and having infinitely many excursions over the local minima. In this case the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables (Ferrari and Gabrielli, Electron. J. Probab. 25, Paper No. 78–1, 2020). We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Gabrielli, Davide. Universita degli Studi dell'Aquila; ItaliaXIII Symposium on Probability and Stochastic ProcessesMéxicoUniversidad Nacional Autónoma de Méxic
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